In practice, the GI2T dispersed fringes are processed following two methods: a)
averaged AC of an interferogram spectral window (
)
for angular diameter determinations, b) averaged CC between two
separate spectral windows of the interferograms (
, 
)
for the analysis of the stellar morphology as a function of Doppler shift across
spectral lines (Fig. 2 (click here)).
In this paragraph, we present theoretical expressions of the standard deviation of
power and cross-power spectra. These spectra are computed by Fourier transforming the
AC/CC of photon events in clipped images. The noise which affects
these spectra comes from both the atmospheric turbulence (speckle noise) and photon
noise.
Roddier & Léna (1984) have given the variance of a photon unbiased estimate of the power
spectrum as:
where N is the average number of photons in the spectral channel and
is the Fourier transform of the instantaneous irradiance
distribution in the image.
Petrov (1986) has described the combined influence of atmospheric fluctuations and
photon noise on the estimation of cross-power spectrum. The variance of this quantity
is defined as:
where N1 and N2 are the average photon number in each spectral channel and
and 
are the Fourier transforms of the instantaneous
irradiance distributions in these channels.
This can be applied to the real and imaginary part of the cross-power
spectrum:
Note that in Eqs. (2) and (3), 
is a
complex quantity.
Figure 2: Numerical simulation of the PCH. From the top to the bottom:
typical phase disturbed wavefronts in the output-pupil plane of the GI2T,
resulting dispersed fringes over 100 Å at the focal plane of the
interferometer (the different spectral windows, used in the AC and CC
computations, are presented), resulting power spectrum to which the PCH
has been added